So I set up my team and it has Dawnstar in. My opponent sets hit team up and it has 4 OOTS Batmen in it. I choose Batman as my Quarry. Does it work against all of them or just 1?

Since it says, "this chraracter" and not "all character's with the same name" or similar wording, I'm with the opinion that it would be that one individual character.

One instance of a character.

I have a question on a variation of this - if I have 2 Dawnstar's on my team, can both choose the same quarry and everyone gets a +2 bonus?

I wouldnt see why not, bob

just remember rule of 3. you can have 4 dawnstars and all chose the same figure, and still get a +3. however, this also stays a +3 even if you get one outwitted, or ko'd, so it has some long term advantages; so if you're fighting one huge tent, all the better, if you're fighting 4 bats vs 4 dawnstars, i feel sorta sorry for reality as you break it with your multi-dimensional war, you heroic fiends!

Remember that the Ro3 is applied to the cumulative total of modifiers when used.

Say that I had a guy performing a Haymaker. He'll get the -1 from that and the +4 from the Dawnstars for a total of +3.

If I played in a game with 4 Dawnstars vs. 4 Batmen, on either side, I think I'd scream from trying to remember who had quarried whom.

Just use a separate color ID ring (or something similar) for each quarry, and you'll be fine.